Atlas › Modular forms › Weight 2 › Level 822 › 822.2.a › 822.2.a.l

Newform orbit 822.2.a.l

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q + 4q^{2} + 4q^{3} + 4q^{4} - 4q^{5} + 4q^{6} + 9q^{7} + 4q^{8} + 4q^{9} - 4q^{10} + 3q^{11} + 4q^{12} + 3q^{13} + 9q^{14} - 4q^{15} + 4q^{16} + 4q^{18} + 8q^{19} - 4q^{20} + 9q^{21} + 3q^{22} - 7q^{23} + 4q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{5} + 3T_{7}$ , whose minimal polynomial is

$x^{4} - 23x^{3} + 70x^{2} + 1319x - 7041$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
3	$-1$
137	$-1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = -1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = +1$ : each L-function in the orbit has even analytic rank.

Traces of Hecke eigenvalues

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97
$\operatorname{tr} a_p$	4	4	−4	9	3	3	0	8	−7	2	−6	16	−7	7	−25	6	−13	−8	−7	−5	3	0	0	−15	−16

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.353 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound. The label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 2, 3, 6, 137, 274, 411, 822]: [0, 1, 1, 3, 12, 36, 48, 142] (certified at ell = 2147483249)
T = [(5, 1), (7, 3)]; h of degree 23 = 1 x 23 (phi(ord eps) x dim S_k^new); 4 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(822, 2)[11]   # 822.2.a.l

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 822:2 1000